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Question

Let the coordinates of the points A,B,C be (1,8,4),(0,11,4) and (2,3,1) respectively. What are the coordinates of the point D which is the foot of the perpendicular from A on BC?

A
(3,4,2)
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B
(4,2,5)
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C
(4,5,2)
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D
(2,4,5)
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Solution

The correct option is C (4,5,2)
given A(1,8,4)
B(0,11,4)
C(2,3,1)
Cartesian Equation of line passing through Pts B and C is given by
xx1x2x1=yy1y2y1=zz1z2z1
Now here substitute values, we get
x020=y+113+11=z414x2=y+118=z43
Now the direction ratio of AL is (xx3),(yy3),(zz3)
The coordinates of the points L on line BC is given by
x=2λ,y=8λ11,z=3λ+4
Directions ratios (2λ1),(8λ19),(3λ)
Since AL is perpendicular to Line BC
a1a2+b1b2+c1c2=0
(2)(2λ1)+8(8λ19)3(3λ)=0
4λ2+64λ152+9λ=0
77λ=154λ=2
Hence coordinates are
(2λ,8λ11,3λ+4)
4,5,2.

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